Massive Cosmological Correlators from Flat Space: a Laplace-Space Approach

Technical Summary: Massive Cosmological Correlators from Flat Space: a Laplace-Space Approach

Problem Statement
Cosmological correlators—the late-time $n$-point functions of primordial perturbations—are the primary observables for probing high-energy physics during inflation. However, their computation is hindered by the breaking of time-translation invariance in an expanding universe. This breaking distorts mode functions from simple plane waves into special functions (Hankel or Whittaker functions) and forces correlators to be expressed as nested time integrals over products of these distorted modes. While various techniques exist (cosmological bootstrap, Mellin-space representations, dispersive methods, family-tree decompositions), handling massive field exchanges remains particularly challenging. Existing methods often struggle to provide closed-form, rapidly convergent analytic expressions valid across the entire kinematic domain, particularly for massive internal lines.

Methodology
The authors develop a new computational framework based on the physical observation that, deep inside the Hubble radius, cosmological modes oscillate as flat-space plane waves, with spacetime curvature only becoming significant as modes stretch toward the horizon. The core of the method involves a Laplace transform applied to the time-dependent building blocks of correlators.

1. Plane-Wave Representation via Laplace Transform:

   • The authors perform a Wick rotation to turn early-time oscillations ($e^{-ik\tau}$) into decaying exponentials.
   • They apply a Laplace transform to the mode functions (viewed as functions of $z = k\tau$).
   • By analyzing the analytic structure of the transform, they identify a branch cut in the complex dual variable $\lambda$ plane, originating from the Bunch-Davies early-time behavior.
   • Inverting the transform by deforming the contour onto this cut yields a representation of the curved-space mode function as a continuous superposition of plane waves ($e^{-i\lambda k\tau}$), weighted by a kernel (the discontinuity across the cut) that encodes the spacetime geometry, field mass, and dynamics.
   • For massive scalars in de Sitter space, this kernel is identified as a Legendre function $P_{i\mu-1/2}(\lambda)$.

2. Reduction to Flat-Space Integrals:

   • In this "Laplace space," every massive internal line in a Feynman diagram is replaced by a plane wave dressed by the Legendre kernel.
   • Consequently, the nested time integrals of the in-in formalism collapse into elementary flat-space (massless) time integrals.
   • The complexity of the curved background is entirely absorbed into the known $\lambda$-space kernels and the integration measure, while the time integration itself becomes a standard flat-space calculation.

3. Diagrammatic Rules:

   • The authors derive a set of diagrammatic rules to construct the Laplace-space integrand directly from a diagram with polynomial interactions.
   • These rules handle Schwinger-Keldysh ($\pm$) vertices, time-ordering, and the specific differential operators required to account for power-law factors in the mode functions (e.g., for conformally coupled external legs).
   • The method is shown to be applicable to tree-level and loop-level diagrams, as well as to fields with chemical potentials (twisted Whittaker functions).

Key Contributions and Results

• Master Series for Single-Exchange Correlators: Applying the formalism to the massive single-exchange correlator, the authors derive a single, closed-form, rapidly convergent series. This series is valid throughout the entire kinematic domain without the need to patch together separate expansions (e.g., for different energy regimes).
• Transparent Singularity Structure: The Laplace representation makes the analytic structure of the correlator transparent "from flat space":
  • Total-Energy Singularities: These arise when the total energy vanishes. In the Laplace picture, this corresponds to a pinch at the early-time corner of the dual integration domain (where $\lambda \to 1$). The coefficient is shown to be independent of the mass, recovering the flat-space scattering amplitude.
  • Partial-Energy Singularities: These occur when the energy flowing into a single vertex vanishes. In the Laplace picture, this corresponds to a pole in the rational part of the integrand reaching the boundary of the integration domain ($\lambda = 1$). The residue is identified as the mass-dependent Legendre kernel, corresponding to the factorization into a lower-point correlator and a lower-point amplitude.
• Generalizability: While detailed for conformally coupled fields exchanging massive scalars in de Sitter, the authors argue the approach extends essentially unchanged to derivative interactions, non-trivial sound speeds, spin-1 fields with chemical potentials, and time-dependent couplings.

Significance
The paper claims that this Laplace-space approach offers a powerful new tool for primordial cosmology. By trading non-trivial time dependence for plane waves, it resolves the computational bottleneck of nested time integrals. The method provides a unified framework that:

1. Handles massive field exchanges, a regime where previous integral transform methods (like Mellin space) have been less effective or confined to massless/conformal cases.
2. Yields analytic results that are both closed-form and rapidly convergent, avoiding the patching of expansions required by other methods.
3. Clarifies the origin of singularities in cosmological correlators by mapping them directly to flat-space data and the geometry of the dual integration domain.

The authors position this work as a complementary approach to the cosmological bootstrap and Mellin-space methods, rooted specifically in the asymptotic early-time symmetry (flat-space behavior) of the modes.